By [1]}, it suffices to show that \(S\)  preserves fibrant objects as a functor \({\Delta {T}\mathrm {op}}\rightarrow {H}_\infty ^{p I}\) .
By Proposition REF  we are thus left to check that, for every \(T \in {\Delta {T}\mathrm {op}}\)  and any \(c \in {{C}\mathrm {art}}\) , the canonical morphism \(S(T)(*) \rightarrow S(T)(c)\)  is a weak equivalence in \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\) .
However, recalling the canonical isomorphism \(S(T)(c) \cong \operatorname{Sing}(T^{\mathrm {D}\underline{c}})\) , this follows readily from the fact that both \(*\)  and \(c\)  are cofibrant in \({\Delta {T}\mathrm {op}}\) , that \(T\)  is fibrant, and that \(\operatorname{Sing}\)  is a right Quillen functor.
Consequently, the functor \(c \mapsto S(T)(c)\)  maps the weak equivalence \(c \rightarrow *\)  in \({\Delta {T}\mathrm {op}}\)  to a weak equivalence in \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\) .
